Why do this problem?
The problem gives practice in using the notation for Binomial
coefficients and manipulating algebraic expressions. In problem
solving mode, if they can't get started, they might first try to
work on the formula for small integer values of $n$.
Possible approach
Use as a revision exercise.
Key questions
If ${2n \choose n}$ is a binomial coefficient in the expansion of
some power of $(1 + x)$ what can you say about the expansion and
about the term where it occurs?
What do we know about ${n\choose r}$ and ${n\choose n-r}$?
Possible support
Ask learners to find the coefficient of $x^2$ in the expansion
of $(1+x)^4$, the coefficient of $x^3$ in the expansion of $(1
+x)^6$ and then the coefficient of $x^4$ in the expansion of $(1 +
x)^8$ and then ask them to try to connect their results to the
problem given.
You could ask students to show that the sum of the $n$th row in
Pascal's Triangle is $2^n$ first - so that they have a sense of
achievement even if they don't succeed in proving the result in
this problem.